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Springer Monographs in Mathematics Filippo Bracci Manuel D Contreras Santiago Díaz-Madrigal Continuous Semigroups of Holomorphic Self-maps of the Unit Disc Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto A Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in ﬁelds of mathematical research that have acquired the maturity needed for such a treatment They are sufﬁciently self-contained to be accessible to more than just the intimate specialists of the subject, and sufﬁciently comprehensive to remain valuable references for many years Besides the current state of knowledge in its ﬁeld, an SMM volume should ideally describe its relevance to and interaction with neighbouring ﬁelds of mathematics, and give pointers to future directions of research More information about this series at http://www.springer.com/series/3733 Filippo Bracci Manuel D Contreras Santiago Díaz-Madrigal • • Continuous Semigroups of Holomorphic Self-maps of the Unit Disc 123 Filippo Bracci Dipartimento di Matematica Università di Roma “Tor Vergata” Roma, Italy Manuel D Contreras Departamento de Matemática Aplicada II and IMUS Universidad de Sevilla Sevilla, Spain Santiago Díaz-Madrigal Departamento de Matemática Aplicada II and IMUS Universidad de Sevilla Sevilla, Spain ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-36781-7 ISBN 978-3-030-36782-4 (eBook) https://doi.org/10.1007/978-3-030-36782-4 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland A Niccolò, che, da grande, prima vuole guidare una ruspa e poi fare matematica Filippo A mis hijos, Carlos y Elena, por todo y por tanto Manolo A Flora, Javi y Sara Santi Acknowledgements It was a sunny and hot day in Nahariya some years ago when we started discussing the idea of writing a book about semigroups of holomorphic self-maps of the unit disc Since the wonderful books on the subject by Marco Abate, Mark Elin, Simeon Reich, and David Shoikhet, there had been no sources in book form containing the various advances of the intervening years, and many colleagues seemed interested in having an updated complete reference source We subsequently worked on the raw material which ﬁnally became the present book During the years needed to see the “light at the end of the tunnel”, we proﬁted from and very much enjoyed discussions with colleagues and friends It is our pleasant duty to thank all of those who helped us Special thanks are due to our friend Pavel “Pasha” Gumenyuk who gave us priceless comments, ideas, and constructive criticisms, besides offering much philosophical advice about the book Certainly this book—and life—would have been very different without his help It was our great privilege to have the opportunity of knowing and proﬁting from the experience, encouragement, and help of our friend Christian Pommerenke, from whom we learned a lot We cannot forget our beloved friend Sasha Vasil’ev, whose constant encouragement was essential to us Wherever you are now, we can imagine you are taking a look at the book with a cerveza in your hand! We wish to thank Hervé Gaussier, from whom we learned a lot about Gromov’s hyperbolicity theory The hours and hours spent at the coffee bar in Seville discussing hyperbolic geometry led us to simplify many proofs in the book, as well as other aspects Thanks are also owed to Andy Zimmer, whose enthusiasm and skill allowed us to understand much better part of the theory We would also like to thank Marco Abate: besides promising to read the book, his work has always been a great inspiration for all of us Leandro Arosio helped us with many interesting comments and clever ideas Thanks! vii viii Acknowledgements We further wish to thank the following great friends, collaborators, and masters, who, in one way or another, contributed a lot to help us: Dimitri Betsakos, Mark Elin, Pietro Poggi-Corradini, Simeon Reich, David Shoikhet, and Aristos Siskakis We also thank Graziano Gentili for his constant support and friendship Filippo wants to thank his parents, Renzo and Anna, for always being there, and his wife Ele for her patience (at least, sometimes), support, and love Manolo wants to express his deepest gratitude to Mara José, his wife, for her constant support and encouragement, especially during the time dedicated to writing this book Life would not be so beautiful without her Thanks! Santi is thankful and deeply indebted to his wife Flora for her endless patience and love Last but not least, we want to express our gratitude to the institutions that have supported us during these years of work: Departamento de Matemática Aplicada II and the Instituto de Matemáticas IMUS, Universidad de Sevilla; Dipartimento di Matematica, Università di Roma “Tor Vergata” (and the related MIUR Excellence Department Project MATH@TOV), and the ERC grant “HEVO” Rome, Italy Seville, Spain October 2019 Filippo Bracci Manuel D Contreras Santiago Díaz-Madrigal Introduction Continuous one-parameter semigroups of holomorphic self-maps of the unit disc D in the complex plane C have been a subject of study since early 1900s, both for their intrinsic interest in complex analysis and for applications In recent years, there has been a lively development of the theory, and deep achievements about boundary behavior and dynamical aspects have been obtained Various key results are contained in different research papers and the aim of this book is to give a systematic and uniﬁed account of the subject, from the very ﬁrst deﬁnitions up to the latest results The theory of local groups (or flows) is intrinsically related to the theory of differential equations Given a vector ﬁeld X deﬁned and smooth on some open set U of Rn (or more generally on a manifold), and a point x0 U, the Cauchy problem @xðtÞ @t ẳ Xxtịị; x0ị ẳ x0 has a unique smooth solution uðx0 ; ÁÞ deﬁned in some interval Ix0 of R containing such that ux0 ; 0ị ẳ x0 The map ðx; tÞ 7! uðx; tÞ is well-deﬁned and smooth on V Â J, where V is a neighborhood of x0 and J is a small interval containing By the uniqueness of solutions of the Cauchy problem, ux; t ỵ sị ẳ uux; tị; sị for all s; t J such that s ỵ t J and uðx; tÞ V The family ðuðÁ; tÞÞ is a one-parameter local group, the flow of the vector ﬁeld X If Ix0 ¼ R for every x0 , then the maps x 7! uðx; tÞ are diffeomorphisms of U for all t R, and ðuðÁ; tÞÞ is a one-parameter group In other words, X deﬁnes a continuous action of the group R on U On the other hand, if Ix0 contains ẵ0; ỵ 1ị for every x0 U, then ðuðÁ; tÞÞ is a one-parameter semigroup and ut :ẳ u; tị is an injective smooth map of U into U for each t ! The semigroup equation, ut ỵ s ẳ ut us , t; s ! implies that un ¼ un ¼ u1 u1 (n times composition of u1 with itself), and the behavior of the orbits of fun g is strictly related to the analytic properties of the vector ﬁeld X, which can be easier to understand Nevertheless, it is often the case that a dynamical system is described by a single self-map f : U ! U, and one is interested in ix x Introduction understanding the asymptotic behavior of the orbits ff n g If f can be embedded into a continuous group (or a semigroup) of functions, the orbits of f can then be studied using vector ﬁelds However, the embedding problem is usually very hard and not always solvable, not even locally In the complex setting, Koenigs [91] was one of the ﬁrst mathematicians to solve the local embedding problem, by ﬁnding solutions to the so-called Schröder equation Namely, he proved that, given a holomorphic self-map f of the unit disc D ﬁxing the origin and whose derivative f ð0Þ ¼ eÀk (with k C, Rek [ 0—and that is the only possibility unless f is linear, by the Schwarz Lemma), one can ﬁnd a holomorphic function u : D ! C, locally invertible at such that ðu f ịzị ẳ ek uzị for all z D Since z 7! eÀkt z, t R, is a group of automorphisms of C, Koenigs’ result implies that f can be locally embedded into a continuous group At the beginning of the twentieth century, Tricomi [124] dealt with problems which, translated into modern language, were related to the asymptotic behavior of continuous one-parameter semigroups of holomorphic self-maps of the unit disc In 1923, Loewner [95] introduced what is nowadays called “Loewner theory” to tackle extremal problems in complex analysis Such a theory, as developed in particular by Pommerenke [102], contains the germ of elliptic semigroups theory and relates semigroups to certain ordinary differential equations Later on, in 1943, Kufarev [94] introduced an ordinary differential equation whose solutions are pretty much related to continuous non-elliptic semigroups In 1939, Wolff [128] studied continuous iteration in the half-plane and proved a type of continuous Denjoy-Wolff theorem, which describes the asymptotic behavior of the trajectories of continuous one-parameter semigroups of holomorphic self-maps of the unit disc In 1968, Karlin and McGregor [85, 86] studied the (global) embedding problem of the probability generating function of a simple discrete time Markov process into a continuous one-parameter semigroup of holomorphic self-maps of the unit disc Such a process, introduced in 1875 by Galton and Watson [126] in connection with the probability of extinction of family names, can be formulated in mathematical terms by means of a discrete dynamical system given by a holomorphic self-map of the unit disc The interest in the Galton-Watson branching process flourished when it was proposed as a basic model for many other physical processes, and this created a need to understand the theory of continuous one-parameter semigroups of holomorphic self-maps of the unit disc In 1978, Berkson and Porta [11] studied continuous one-parameter semigroups of holomorphic self-maps of the unit disc in connection with composition operators They proved the existence of the inﬁnitesimal generator and what is now referred to as the Berkson-Porta Formula In 1981, Heins [82] investigated continuous one-parameter semigroups of holomorphic self-maps of Riemann surfaces and proved the existence of the Koenigs function for semigroups of the unit disc Since then, many papers and books focusing on different aspects of this theory have been written We cite here the book by Abate [1], which mainly focuses on the asymptotic behavior of orbits of discrete dynamical systems and contains a chapter 18.3 Exceptional Maximal Contact Arcs and the Denjoy-Wolff Point 551 Example 18.3.5 Let Ω = {z ∈ S : Im z Re z > −1} and let h : D → Ω be a Riemann map of Ω For t ≥ 0, let φ˜ t (z) := h −1 (h(z) + it) Clearly, (φ˜ t ) is a semigroup in D The point σ = limt→0 h −1 (t − i/2t) is a boundary super-repelling fixed point of third type for this semigroup (see Example 14.5.4) Let v : (0, 1) → R be a continuous function and define g(z) := Re z + i(Im z + v(Re z)) Write Ωv := g(Ω) = {z ∈ S : Im z > − Re1 z + v(Re z)} Let h v : D → Ωv be a Riemann map of Ωv Let φt (z) := h −1 v (h v (z) + it), z ∈ D and t ≥ The semigroup (φt ) is clearly topologically conjugated to (φ˜ t ) On the one hand, taking v(x) := 2/x, we obtain a semigroup whose unique fixed point is its Denjoy-Wolff point and the map f = h −1 v ◦g◦h sends the exceptional maximal contact arc starting at σ to the Denjoy-Wolff point of (φt ) In particular, the boundary super-repelling fixed point of third type σ is sent to the Denjoy-Wolff point On the other hand, taking v(x) = 1/x we obtain a semigroup whose unique fixed point is the Denjoy-Wolff point and the map f = h −1 v ◦g◦h sends the exceptional maximal contact arc starting at σ onto an exceptional maximal contact arc for (φt ) having a non-fixed point as initial point In particular, the boundary super-repelling fixed point of third type σ is mapped to a contact, not fixed, point Proposition 18.3.6 Let (φt ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D Let M be an exceptional maximal contact arc for (φt ) whose initial point is σ ∈ ∂D (1) If σ is a repelling fixed point for (φt ) and (φ˜ t ) is a hyperbolic semigroup of holomorphic self-maps of D, topologically conjugated to (φt ) via the homeomorphism f : D → D, then (φ˜ t ) has an exceptional maximal contact arc whose initial point ∠ lim z→σ f (z) is a repelling fixed point (2) If σ is not a boundary regular fixed point for (φt ), then there exists a homeomorphism f : D → D such that (φ˜ t ) := ( f −1 ◦ φt ◦ f ) is a hyperbolic semigroup of holomorphic self-maps of D and lim z→ p f (z) = τ for all p ∈ M Proof (1) By Remark 18.3.2, f (σ ) := ∠ lim z→σ f (z) is a repelling fixed point of (φ˜ t ) If f (σ ) does not belong to the closure of an exceptional maximal contact arc for (φ˜ t ), by Theorem 18.1.3, f −1 has unrestricted limit at f (σ ) and σ = lim (0,1) r →1 f −1 ( f (r σ )) = f −1 ( f (σ )) ∈ / E(φt ), a contradiction Therefore f (σ ) belongs to the closure of an exceptional maximal contact arc M for (φ˜ t ) Being f (σ ) a fixed point, it cannot sit in M, and, by Proposition 14.2.6, is in fact the initial point of M (2) Let (Ω = (0, ρ) × R, h, z → z + it), < ρ < +∞, be the holomorphic model of (φt ) and let Q := h(D) Let M be an exceptional maximal contact arc with initial point σ By Theorem 14.2.10, we can assume without loss of generality that Re h(z) = for all z ∈ M Assume σ is not a boundary regular fixed point for (φt ) 552 18 Topological Invariants If σ is a super-repelling boundary fixed point for (φt ), let / Q, s ∈ (0, 1/n]} λn = sup{y : s + i y ∈ Since σ is super-repelling, by Corollary 13.6.7, we have that limr →1 Re h(r σ ) = and limr →1 Im h(r σ ) = −∞ Thus λn tends to −∞ For each n ∈ N, take < sn ≤ / Q and yn ≥ λn − 1/n Take a monotone subsequence 1/n and yn such that sn + i yn ∈ (sn k ) of (sn ) and a continuous function v : (0, ρ) → R such that v(sn k ) = −2λn k Define g : Ω → Ω by g(x + i y) := x + i(y + v(x)) Note that g is a homeomorphism Let h : D → g(Q) be a Riemann map of g(Q) Let (φ˜ t ) be the semigroup ˜ in D defined by φ˜ t (z) := h −1 (h (z) + it) Clearly, (φt ) is topologically conjugated / g(Q) and to (φt ) Since wn = g(sn + i yn ) = g(sn ) + i yn ∈ lim Im (wn k ) = lim(yn k + v(sn k )) ≥ lim(−λn k − 1/n k ) = +∞, k k k it follows that f maps the exceptional maximal contact arc M to the Denjoy-Wolff point of (φ˜ t ) Next, suppose that σ is not a boundary fixed point Then, by Theorem 11.1.4, there exists limr →1 h(r σ ) = i y0 with y0 ∈ R Therefore there exist points xn + i yn ∈ Ω \ Q such xn goes to and yn goes to y0 Take a continuous function v : (0, ρ) → R such that lim x→0 v(x) = +∞ As before, define g : Ω → Ω by g(x + i y) := x + i(y + v(x)) and let h : D → g(Q) be any Riemann map By construction, g(xn + i yn ) ∈ Ω \ h (D) and Im g(xn + i yn ) goes to +∞ Therefore the semigroup (φ˜ t ) defined by φ˜ t (z) := h −1 (h (z) + it) for t ≥ is topologically conjugated to (φt ) and f maps the exceptional maximal contact arc M to the Denjoy-Wolff point of (φ˜ t ) The cluster set of f at an exceptional maximal contact arc is described by the following proposition: Proposition 18.3.7 Let (φt ) and (φ˜ t ) be two non-elliptic semigroups in D with holomorphic models (Ω1 = I1 + iR, h , z → z + it) and (Ω2 = I2 + iR, h , z → z + it), respectively Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D Let τ and τ˜ be the Denjoy-Wolff points of (φt ) and (φ˜ t ), respectively Assume that M is an exceptional maximal contact arc for (φt ) Denote by S = {z ∈ D \ {0} : z/|z| ∈ M} and let σ be the initial point of M Then the set E(M) = w ∈ D : ∃{z n } ⊂ S, z n → z ∈ M, f (z n ) → w is a compact connected arc in ∂D containing τ˜ which is either equal to {τ˜ } or it is contained in the closure of an exceptional maximal contact arc for (φ˜ t ) Proof Let Q := h (D) and let Q := h (D) By Lemma 18.1.2, (φt ) and (φ˜ t ) are topologically conjugated if and only if there exists a homeomorphism g : Ω1 → Ω2 given by (18.1.1) and g(Q ) = Q By Proposition 9.7.3, f = h −1 ◦ g ◦ h1 18.3 Exceptional Maximal Contact Arcs and the Denjoy-Wolff Point 553 Since ∠ lim z→τ f (z) = τ˜ by Proposition 18.3.1, it follows that τ˜ ∈ E(M) Moreover, it is easy to see that E(M) is a compact subset of ∂D Therefore we are left to check that E(M) is connected Indeed, assume this is not the case Then there exist two compact sets A and B such that E(M) = A1 ∪ A2 and A1 ∩ A2 = ∅ Denote by k > the euclidean distance between A and B For j = 1, 2, take w j ∈ A j , z n, j ∈ D, z for all n, with |zn,n, jj | ∈ M, {z n, j } → z j ∈ M and { f (z n, j )} → w j We may assume that | f (z n, j ) − w j | < k/4 for all j and n Let Cn be the arc in M that joins z n,2 and |z n,2 | Γn = [z n,1 , rn z n,1 ] ∪ rn Cn ∪ [z n,2 , rn z n,2 ] z n,1 |z n,1 | with where rn = max{|z n,1 |, |z n,2 |} Notice that Γn is connected Consider the continuous function l : Γn → R being l(z) the distance between f (z) and A1 Since l(z n,1 ) < k/4 and w2 ∈ A2 , we have l(z n,2 ) ≥ k − |z n,2 − w2 | ≥ k − k/4 = 3k/4 Thus there is αn ∈ Γn such that l(αn ) = k/2 Since |αn | ≥ min{|z n,1 |, |z n,2 |}, we can take a subsequence such that αn k → z ∈ M and f (αn k ) → w Clearly, w ∈ E(M) and l(w) = k/2 A contradiction Hence E(M) is connected Finally, if E(M) = {τ˜ }, E(M) is contained in the closure of an exceptional maximal contact arc for (φ˜ t ), for otherwise f −1 would map points of ∂D \ E(φ˜ t ) into E(φt ) Example 18.3.8 Let Ω1 = {z ∈ S : Im z > 0} and h : D → Ω1 a Riemann map of Ω1 Consider the semigroup (φt ) defined by φt (z) := h −1 (h (z) + it), t ≥ The arc M = h −1 ([0, ∞)i) is an exceptional maximal contact arc for (φt ) Define g : S → S as g(x + i y) := x + i(y − 1/x), Ω2 := g(Ω1 ) = {z ∈ S : Im z Re z > −1} and let h : D → Ω2 be a Riemann map of Ω2 The semigroup (φ˜ t ) defined by φ˜ t (z) := h −1 (h (z) + it), t ≥ 0, is topological conjugated to (φt ) via the homeomorphism ˜ ˜ f = h −1 ◦ g ◦ h The semigroup (φt ) has an exceptional maximal contact arc M = −1 h (Ri) with initial point a fixed point σ Notice that for all w ∈ M it holds σ = lim z→w f (z), namely, the map f sends the arc M to the point σ This does not contradict the previous proposition, since M˜ is the cluster set of f at the DenjoyWolff point of (φt ) 18.4 Elliptic Case In this final section we show how to recover the results of Sects 18.1 and 18.2 in case of elliptic semigroups which are not groups The key point is to replace Lemma 18.1.2 by the following lemma (whose proof is similar to that of Lemma 18.1.2): Lemma 18.4.1 Let (φt ) and (φ˜ t ) be two elliptic semigroups in D, which are not groups Let (C, h , z → eλ1 t z) be a holomorphic model of (φt ) and let (C, h , z → eλ2 t z) be a holomorphic model of (φ˜ t ) Then, (φt ) and (φ˜ t ) are topologically conjugated if and only if there exist a homeomorphism u of the unit circle ∂D ◦ g0 ◦ θλ1 satisfies and a continuous map v : ∂D → (0, +∞) such that g := θλ−1 554 18 Topological Invariants g(h (D)) = h (D), where g0 (z) := 0, z=0 , z ∈ C, |z|u(z/|z|)v(z/|z|), z = and, given λ = a + ib with a < 0, b θλ (z) := z|z|−(1+1/a) exp −i Log|z| , z ∈ C, z = 0, and θλ (0) := a As already remarked, if the semigroup (φt ) is elliptic, the set E(φt ) = ∅ Using the previous lemma and mimicking the proof of Theorem 18.1.3, one can prove the following extension result: Theorem 18.4.2 Let (φt ) and (φ˜ t ) be two elliptic semigroups in D, which are not groups Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D Then f extends to a homeomorphism f : D → D Moreover, for all σ ∈ ∂D the life-time T (σ ) = T ( f (σ )) and f (φt (σ )) = φ˜ t ( f (σ )) for all t ≥ And also, Proposition 18.4.3 Let (φt ) and (φ˜ t ) be two elliptic semigroups in D, which are not groups Suppose (φt ) and (φ˜ t ) are topologically conjugated via the homeomorphism f : D → D (1) If M is a maximal contact arc for (φt ), then f extends to a homeomorphism from M onto f (M) and f (M) is a maximal contact arc for (φ˜ t ) (2) Let σ ∈ ∂D be a boundary fixed point for (φt ) Then the unrestricted limit f (σ ) := lim f (z) ∈ ∂D z→σ exists and f (σ ) ∈ ∂D is a boundary fixed point for (φ˜ t ) Moreover, (i) if σ is a boundary regular fixed point for (φt ), then f (σ ) is a boundary regular fixed point for (φ˜ t ); (ii) if σ is a boundary super-repelling fixed point of first type (respectively of second type) for (φt ), then f (σ ) is a boundary super-repelling fixed point of first type (respectively of second type) for (φ˜ t ) We end this section showing that Theorem 18.4.2 is no longer true for groups of elliptic automorphisms 18.4 Elliptic Case 555 Example 18.4.4 Consider the group of automorphisms (φt ) where φt (z) = eit z for all t ∈ R and z ∈ D and the continuous function f : D → D given by f (z) = z exp(i ln(1 − |z|)) for all z ∈ D It is clear that f is a homeomorphism of the unit disc with inverse function f −1 (z) = z exp(−i ln(1 − |z|)) and f (φt (z)) = φt ( f (z)), t ≥ 0, z ∈ D However, f has no continuous extension at any point of ∂D 18.5 Notes The chapter is based on [30] References Abate, M.: Iteration Theory of Holomorphic Maps on Taut Manifolds Mediterranean Press, Rende (1989) Aharonov, D., Elin, M., Reich, S., Shoikhet, D.: Parametric representation of semi-complete vector fields on the unit balls in Cn and in Hilbert space Rend Mat Acc Lincei 10, 229–253 (1999) Ahlfors, L.V.: Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd edn McGraw-Hill, New York, NY (1979) Arendt, W., Chalendar, I.: Generators of semigroups on Banach spaces inducing holomorphic semiflows Israel J Math 229, 165–179 (2019) Arosio, L.: The stable subset of a univalent self-map Math Z 281, 1089–1110 (2015) Arosio, L.: Canonical models for the forward 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Mathematics, Cambridge University Press (1991) 128 Wolff, J.: L’équation différentielle dz/dt = w(z) =fonction holomprphe partie réelle positive dans un demi-plan Compos Math 6, 296–304 (1939) Index A Accessible point via a Jordan arc , 113 Accessible prime end, 113 Algebraic group, 222 Algebraic semigroup, 214, 222 Angular derivative, 38 Arc length parameter of a curve, 120 Area Theorem, 80 Argument of a complex number, xxiii B Backward invariant set of a semigroup, 372 Backward orbit, 355 Base space of a holomorphic semi-model, 236 Berkson-Porta Formula, 278 β-point, 431 Bloch function, 187 Bloch space B, 187 Boundary critical point, 330 Boundary dilation coefficient, 18 Boundary fixed point of a self-map of the unit disc, 54 Boundary fixed point of a semigroup, 327 Boundary regular critical point, 330 Boundary regular fixed point of a self-map, 55 Boundary regular fixed point of a semigroup, 329 Carathéodory Kernel Convergence Theorem, 89 Carathéodory topology, 101 Cauchy’s Functional Equation, 208 Cayley transform, Circle, xxiv Circular null chain, 93 Cluster set of a curve, 55 Cluster set of a function, 108 Complete vector field, 274 Conjugated semigroups, 267 Connected im kleinen, 99 Contact arc of a semigroup, 411 Contact point of a self-map of the unit disc, 54 Contact point of a semigroup, 411 Continuous group, 222 Continuous semigroup, 205, 222 Convex domains, 127 Cross cut, 91 D Denjoy-Wolff point, 51 Denjoy-Wolff point of the semigroup, 220 Disc, xxiv Divergence rate, 231 Domain, xxiv Dual infinitesimal generator, 446 E Elliptic function, 51 C Elliptic groups, 218 Canonical model, 248 Elliptic semigroups, 220 Carathéodory boundary of a simply connected domain, 97 Embedding, 262 © Springer Nature Switzerland AG 2020 F Bracci et al., Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-36782-4 563 564 End points of a cross cut, 91 End points of an open arc, xxiv Euclidean diameter diamE , 73 Exceptional regular backward orbit, 361 F Final point of a contact arc, 411 Finite contact point of a holomorphic function with non-negative real part, 64 Fixed point of a semigroup, 327 Fixed points of a holomorphic self-map, 54 G Geodesics in simply connected domains, 122 Green function, 286 Group in a Riemann surface, 222 Group in D, 214 H Hardy-Littlewood maximal function, 32 Harmonic measure in D, 172 Harmonic measure in simply connected domains, 177 Herglotz representation, 59 Holomorphically conjugated semigroups, 267 Holomorphic conjugation of holomorphic models, 267 Holomorphic model, 236 Holomorphic semi-model, 235 Horizontal sector, 135 Horocycle, 17 Horocycle in a simply connected domain, 153 Hyperbolic distance of points in D, 11 Hyperbolic distance on Riemann surfaces, 14 Hyperbolic function, 51 Hyperbolic groups, 218 Hyperbolic length of a curve, 11 Hyperbolic length of a curve in a simply connected domain, 119 Hyperbolic length of Lipschitz curves, 137 Hyperbolic metric, 10 Hyperbolic metric in a simply connected domain, 117 Hyperbolic norm of a vector in a simply connected domain, 118 Hyperbolic petal of a semigroup, 377 Index Hyperbolic projection of a point onto a geodesic, 156 Hyperbolic pseudo-distance, 282 Hyperbolic sector around a geodesic, 135 Hyperbolic semigroups, 220 Hyperbolic step of a regular backward orbit, 356 Hyperbolic steps of order u, 231 I Infinitesimal generator, 275 Abate’s Formula, 286 Ahoronov-Elin-Reich-Shoikhet’s Formula, 288 characterizations, 275, 280, 284, 288 of a group, 289 of a semigroup of linear fractional maps, 292 Inner fixed point of a semigroup, 327 Interior part of a null chain, 92 Intertwining mapping, 236 Isolated radial slit, 435 Isolated spiral slit, 435 Isomorphism of topological models, 269 Iterate of a semigroup in a Riemann surface, 222 Iterate of a semigroup in D, 205 J Jordan arc, 73 Jordan curve, 73 Jordan domain, 105 K Kernel convergence of a sequence of domains, 87 Kernel of a sequence of domains, 87 Koebe arcs, 73 Koebe domain, 155 Koebe’s Distortion Theorem, 83 Koebe 1/4-Theorem, 82 Koenigs function, 248 uniqueness, 248 Kolmogorov’s Backward Equation, 276 L Life-time of a boundary point, 409 Linear fractional map, Index Locally arcwise connected space, 104 Locally connected space, 99 M Maximal half-plane, 389 Maximal invariant curve, 372 Maximal spirallike sector, 387 Maximal strip, 387 Möbius transformation, Multiplier of an inner fixed point, 54 Multiplier of a self-map at a boundary point, 55 N Non-elliptic groups, 218 Non-elliptic semigroups, 220 Non-exceptional regular backward orbit, 361 Non-tangential cluster set of a function, 108 Non-tangential converges to ∞ in H, 45 Non-tangential limit of a function f : D → C, 22 Non-tangential limit of a function f : H → C at ∞, 45 Non-tangential limit of a sequence, 22 Non-tangential maximal function, 31 Null chain, 92 O Open arc, xxiv Orbit of a semigroup, 206 Orthogonal speed of a curve in the disc, 157 Orthogonal speed of a non-elliptic semigroup, 455 P Parabolic function, 51 Parabolic groups, 218 Parabolic petal of a semigroup, 377 Parabolic semigroups, 220 Petal of a semigroup, 375 Poisson integral, 28 Poisson kernel, 18, 280 Pole of a infinitesimal generator, 429 Positive hyperbolic step, 245 Prime end, 97 Prime end impression, 97 Principal part of a prime end, 110 Product Formula, 303 565 Q Quasi-geodesics, 137 Question of embedding, 262 R Radial cluster set, 108 Radial cluster set of a function, 108 Radial limit of a function f : D → C, 22 Reflection through a line, 134 Regular backward orbit, 356 Regular contact point of a self-map, 55 Regular finite contact point of a holomorphic function with non-negative real part, 64 Regular pole of a infinitesimal generator, 429 Regular zero, 338 Regular zeros of a holomorphic function with non-negative real part, 64 Repelling fixed point of the semigroup, 329 Repelling spectral value of a semigroup at a boundary fixed point, 329 Riemann sphere, Riemann surface, S Semicomplete vector field, 273 complete, 274 Semi-conformality at a boundary point, 364 Semi-conjugation map, 234 Semigroup, 206 algebraic in the unit disc, 205 characterizations, 211, 213, 275 elliptic, 220 hyperbolic, 220 in D, 206 iterate, 205 non-elliptic, 220 of hyperbolic rotations, 206 of positive hyperbolic type, 245 of zero hyperbolic type, 245 parabolic, 220 semi-conjugated, 234 trivial, 206 Semigroup of automorphic type, 245 Semigroup of non-automorphism type, 245 Semigroups of linear fractional maps, 226 Semi-strip of width R and height M, 167 Shift of a semigroup, 529 Simple boundary point, 105 Slope of a curve, 501 Spectral value of the semigroup, 220 Spherical diameter, 566 Spherical diameter d S , Spherical distance, Spirallike, 251, 253 Spirallike argument, 390 Spirallike sector, 385 Starlike, 251 Starlike at infinity, 256, 257 Starting point of a contact arc, 411 Starting point of a maximal invariant curve, 372 Stolz region, 23 Strip, 162 Subadditive, 230 Super-repelling fixed point of the first type, 424 Super-repelling fixed point of the second type, 424 Super-repelling fixed point of the semigroup, 329 Super-repelling fixed point of the third type, 424 Symmetry of a simply connected domain with respect to a line, 134 Index T Tangential speed of a curve in a simply connected domain, 157 Tangential speed of a non-elliptic semigroup, 455 Tip of an isolated spiral or radial slit, 435 Topological conjugation of topological models, 270 Topologically conjugated semigroups, 270 Total speed of a non-elliptic semigroup, 454 Trajectory of a semigroup, 206 U Univalent, Unrestricted limit of a function f : D → C, 23 Z Zero hyperbolic step, 245 ... center z0 and radius r is ff C : jf À z0 j
g The unit disc, i.e ., the disc of center and radius is denoted by D The circle of center z0 and radius r [ is just the boundary of the disc with the same... purpose, the book is self- contained and all non-standard (and, mostly, all standard) results are proved in detail The core of the book is Part II, starting at Chap The reader, especially the expert,... behavior of the trajectories of continuous one-parameter semigroups of holomorphic self- maps of the unit disc In 196 8, Karlin and McGregor [8 5, 86] studied the (global) embedding problem of the probability
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